Abstract
Let $V=\mathbb{C}^n$ be endowed with an orthogonal form and $G=\mathrm{O}(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\mathrm{End}_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\mathrm{GL}(V)$, for $r\geq n+1$, $\nu$ has a kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\mathrm{O}(V)$ in $V^{\otimes r}$. We also show how our results extend to the case where $\mathbb{C}$ is replaced by an appropriate field of positive characteristic, and we comment on quantum analogues of our results.
-
[ABP]
M. Atiyah, R. Bott, and V. K. Patodi, "On the heat equation and the index theorem," Invent. Math., vol. 19, pp. 279-330, 1973.
@article {ABP, MRKEY = {0650828},
AUTHOR = {Atiyah, M. and Bott, R. and Patodi, V. K.},
TITLE = {On the heat equation and the index theorem},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {19},
YEAR = {1973},
PAGES = {279--330},
ISSN = {0020-9910},
MRCLASS = {58G10 (35K05)},
MRNUMBER = {0650828},
MRREVIEWER = {Authors' review},
ZBLNUMBER = {0257.58008},
DOI = {10.1007/BF01425417},
} -
[BW]
J. S. Birman and H. Wenzl, "Braids, link polynomials and a new algebra," Trans. Amer. Math. Soc., vol. 313, iss. 1, pp. 249-273, 1989.
@article {BW, MRKEY = {0992598},
AUTHOR = {Birman, Joan S. and Wenzl, Hans},
TITLE = {Braids, link polynomials and a new algebra},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {313},
YEAR = {1989},
NUMBER = {1},
PAGES = {249--273},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {57M25 (17B99 20F36)},
MRNUMBER = {0992598},
MRREVIEWER = {Mark E. Kidwell},
DOI = {10.2307/2001074},
ZBLNUMBER = {0684.57004},
} -
[Br]
R. Brauer, "On algebras which are connected with the semisimple continuous groups," Ann. of Math., vol. 38, iss. 4, pp. 857-872, 1937.
@article {Br, MRKEY = {1503378},
AUTHOR = {Brauer, Richard},
TITLE = {On algebras which are connected with the semisimple continuous groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {38},
YEAR = {1937},
NUMBER = {4},
PAGES = {857--872},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {Contributed Item},
MRNUMBER = {1503378},
DOI = {10.2307/1968843},
ZBLNUMBER = {0017.39105},
} -
[dCP]
C. de Concini and C. Procesi, "A characteristic free approach to invariant theory," Advances in Math., vol. 21, iss. 3, pp. 330-354, 1976.
@article {dCP, MRKEY = {0422314},
AUTHOR = {de Concini, C. and Procesi, C.},
TITLE = {A characteristic free approach to invariant theory},
JOURNAL = {Advances in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {21},
YEAR = {1976},
NUMBER = {3},
PAGES = {330--354},
ISSN = {0001-8708},
MRCLASS = {15A72 (14L99)},
MRNUMBER = {0422314},
MRREVIEWER = {V. L. Popov},
ZBLNUMBER = {0347.20025},
DOI = {10.1016/S0001-8708(76)80003-5},
} -
[DHW]
W. F. Doran IV, D. B. Wales, and P. J. Hanlon, "On the semisimplicity of the Brauer centralizer algebras," J. Algebra, vol. 211, iss. 2, pp. 647-685, 1999.
@article {DHW, MRKEY = {1666664},
AUTHOR = {Doran, IV, William F. and Wales, David B. and Hanlon, Philip J.},
TITLE = {On the semisimplicity of the {B}rauer centralizer algebras},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {211},
YEAR = {1999},
NUMBER = {2},
PAGES = {647--685},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {16G99 (20G05)},
MRNUMBER = {1666664},
MRREVIEWER = {Fabio Gavarini},
DOI = {10.1006/jabr.1998.7592},
ZBLNUMBER = {0944.16002},
} -
[GL96]
J. J. Graham and G. I. Lehrer, "Cellular algebras," Invent. Math., vol. 123, iss. 1, pp. 1-34, 1996.
@article {GL96, MRKEY = {1376244},
AUTHOR = {Graham, J. J. and Lehrer, G. I.},
TITLE = {Cellular algebras},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {123},
YEAR = {1996},
NUMBER = {1},
PAGES = {1--34},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {20C99},
MRNUMBER = {1376244},
DOI = {10.1007/BF01232365},
ZBLNUMBER = {0853.20029},
} -
[GL03]
J. J. Graham and G. I. Lehrer, "Diagram algebras, Hecke algebras and decomposition numbers at roots of unity," Ann. Sci. École Norm. Sup., vol. 36, iss. 4, pp. 479-524, 2003.
@article {GL03, MRKEY = {2013924},
AUTHOR = {Graham, J. J. and Lehrer, G. I.},
TITLE = {Diagram algebras, {H}ecke algebras and decomposition numbers at roots of unity},
JOURNAL = {Ann. Sci. École Norm. Sup.},
FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
VOLUME = {36},
YEAR = {2003},
NUMBER = {4},
PAGES = {479--524},
ISSN = {0012-9593},
CODEN = {ASENAH},
MRCLASS = {20C08 (20F36)},
MRNUMBER = {2013924},
MRREVIEWER = {Richard M. Green},
DOI = {10.1016/S0012-9593(03)00020-X},
ZBLNUMBER = {1062.20003},
} -
[GL04] J. J. Graham and G. I. Lehrer, "Cellular algebras and diagram algebras in representation theory," in Representation Theory of Algebraic Groups and Quantum Groups, Tokyo: Math. Soc. Japan, 2004, vol. 40, pp. 141-173.
@incollection {GL04, MRKEY = {2074593},
AUTHOR = {Graham, J. J. and Lehrer, G. I.},
TITLE = {Cellular algebras and diagram algebras in representation theory},
BOOKTITLE = {Representation Theory of Algebraic Groups and Quantum Groups},
SERIES = {Adv. Stud. Pure Math.},
VOLUME = {40},
PAGES = {141--173},
PUBLISHER = {Math. Soc. Japan},
ADDRESS = {Tokyo},
YEAR = {2004},
MRCLASS = {20C08 (20F36)},
MRNUMBER = {2074593},
MRREVIEWER = {Andrew Mathas},
ZBLNUMBER = {1135.20302},
} -
[HX]
J. Hu and Z. Xiao, "On tensor spaces for Birman-Murakami-Wenzl algebras," J. Algebra, vol. 324, iss. 10, pp. 2893-2922, 2010.
@article {HX, MRKEY = {2725207},
AUTHOR = {Hu, Jun and Xiao, Zhankui},
TITLE = {On tensor spaces for {B}irman-{M}urakami-{W}enzl algebras},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {324},
YEAR = {2010},
NUMBER = {10},
PAGES = {2893--2922},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {17B37 (16S99 20C08 20G05)},
MRNUMBER = {2725207},
MRREVIEWER = {Simon Riche},
DOI = {10.1016/j.jalgebra.2010.08.017},
ZBLNUMBER = {05840426},
} -
[Jo]
V. F. R. Jones, "Hecke algebra representations of braid groups and link polynomials," Ann. of Math., vol. 126, iss. 2, pp. 335-388, 1987.
@article {Jo, MRKEY = {0908150},
AUTHOR = {Jones, V. F. R.},
TITLE = {Hecke algebra representations of braid groups and link polynomials},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {126},
YEAR = {1987},
NUMBER = {2},
PAGES = {335--388},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {46L99 (20F36 22D25 46L35 46L55 57M25)},
MRNUMBER = {0908150},
MRREVIEWER = {Pierre de la Harpe},
DOI = {10.2307/1971403},
ZBLNUMBER = {0631.57005},
} -
[Lo] J. Loday, Cyclic Homology, Second ed., New York: Springer-Verlag, 1998, vol. 301.
@book {Lo, MRKEY = {1600246},
AUTHOR = {Loday, Jean-Louis},
TITLE = {Cyclic Homology},
SERIES = {Grundlhern Math. Wissen.},
VOLUME = {301},
EDITION = {Second},
NOTE = {Appendix E by Mar{\'ı}a O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1998},
PAGES = {xx+513},
ISBN = {3-540-63074-0},
MRCLASS = {16E40 (13D03 17B55 18G60 19D55)},
MRNUMBER = {1600246},
ZBLNUMBER = {0928.19001},
ZBLNUMBER = {0885.18007},
} -
[LZ1]
G. I. Lehrer and R. B. Zhang, "Strongly multiplicity free modules for Lie algebras and quantum groups," J. Algebra, vol. 306, iss. 1, pp. 138-174, 2006.
@article {LZ1, MRKEY = {2271576},
AUTHOR = {Lehrer, G. I. and Zhang, R. B.},
TITLE = {Strongly multiplicity free modules for {L}ie algebras and quantum groups},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {306},
YEAR = {2006},
NUMBER = {1},
PAGES = {138--174},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {17B10 (17B37 20G42)},
MRNUMBER = {2271576},
MRREVIEWER = {Jan E. Grabowski},
DOI = {10.1016/j.jalgebra.2006.03.043},
ZBLNUMBER = {1169.17003},
} -
[LZ2]
G. I. Lehrer and R. B. Zhang, "A Temperley-Lieb analogue for the BMV algebra," in Representation Theory of Algebraic Groups and Quantum Groups, New York: Springer-Verlag, 2010, vol. 284, pp. 155-190.
@incollection {LZ2, MRKEY = {2761939},
AUTHOR = {Lehrer, G. I. and Zhang, R. B.},
TITLE = {A {T}emperley-{L}ieb analogue for the {BMV} algebra},
BOOKTITLE = {Representation Theory of Algebraic Groups and Quantum Groups},
SERIES = {Progr. Math.},
VOLUME = {284},
PAGES = {155--190},
PUBLISHER = {Springer-Verlag},
YEAR = {2010},
MRCLASS = {16T20 (20C08)},
MRNUMBER = {2761939},
MRREVIEWER = {Oleg V. Ogievetsky},
DOI = {10.1007/978-0-8176-4697-4_7},
ADDRESS = {New York},
ZBLNUMBER = {05919683},
} -
[LZ3]
G. I. Lehrer and R. Zhang, "On endomorphisms of quantum tensor space," Lett. Math. Phys., vol. 86, iss. 2-3, pp. 209-227, 2008.
@article {LZ3, MRKEY = {2465755},
AUTHOR = {Lehrer, Gustav Isaac and Zhang, Ruibin},
TITLE = {On endomorphisms of quantum tensor space},
JOURNAL = {Lett. Math. Phys.},
FJOURNAL = {Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics},
VOLUME = {86},
YEAR = {2008},
NUMBER = {2-3},
PAGES = {209--227},
ISSN = {0377-9017},
CODEN = {LMPHDY},
MRCLASS = {17B37 (20C08)},
MRNUMBER = {2465755},
MRREVIEWER = {Vladislav K. Kharchenko},
DOI = {10.1007/s11005-008-0284-1},
ZBLNUMBER = {1213.17015},
} -
[LZZ]
G. I. Lehrer, H. Zhang, and R. B. Zhang, "A quantum analogue of the first fundamental theorem of classical invariant theory," Comm. Math. Phys., vol. 301, iss. 1, pp. 131-174, 2011.
@article {LZZ, MRKEY = {2753673},
AUTHOR = {Lehrer, Gustav Isaac and Zhang, Hechun and Zhang, R. B.},
TITLE = {A quantum analogue of the first fundamental theorem of classical invariant theory},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {301},
YEAR = {2011},
NUMBER = {1},
PAGES = {131--174},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {17B37},
MRNUMBER = {2753673},
MRREVIEWER = {Nicol{á}s Andruskiewitsch},
DOI = {10.1007/s00220-010-1143-3},
ZBLNUMBER = {05843865},
} -
[Lu] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993, vol. 110.
@book {Lu, MRKEY = {1227098},
AUTHOR = {Lusztig, George},
TITLE = {Introduction to Quantum Groups},
SERIES = {Progress in Mathematics},
VOLUME = {110},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1993},
PAGES = {xii+341},
ISBN = {0-8176-3712-5},
MRCLASS = {17B37 (16W30 17-02 17B35 81R50)},
MRNUMBER = {1227098},
MRREVIEWER = {Jie Du},
ZBLNUMBER = {0788.17010},
} -
[P1] C. Procesi, "150 years of invariant theory," in The Heritage of Emmy Noether, Ramat Gan: Bar-Ilan Univ., 1999, vol. 12, pp. 5-21.
@incollection {P1, MRKEY = {1665432},
AUTHOR = {Procesi, Claudio},
TITLE = {150 years of invariant theory},
BOOKTITLE = {The Heritage of {E}mmy {N}oether},
VENUE={{R}amat-{G}an, 1996},
SERIES = {Israel Math. Conf. Proc.},
VOLUME = {12},
PAGES = {5--21},
PUBLISHER = {Bar-Ilan Univ.},
ADDRESS = {Ramat Gan},
YEAR = {1999},
MRCLASS = {01A60 (01A50 13-03)},
MRNUMBER = {1665432},
MRREVIEWER = {C. R. Fletcher},
ZBLNUMBER = {0931.15021},
} -
[P2] C. Procesi, Lie Groups. An Approach Through Invariants and Representations, New York: Springer-Verlag, 2007.
@book {P2, MRKEY = {2265844},
AUTHOR = {Procesi, Claudio},
TITLE = {Lie Groups. An Approach Through Invariants and Representations},
SERIES = {Universitext},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2007},
PAGES = {xxiv+596},
ISBN = {978-0-387-26040-2; 0-387-26040-4},
MRCLASS = {22E10 (05E10 05E15 14L24 14M15 17B10 20G05 22-02)},
MRNUMBER = {2265844},
MRREVIEWER = {James E. Humphreys},
ZBLNUMBER = {1154.22001},
} -
[P3] C. Procesi, "The invariant theory of $n\times n$ matrices," Advances in Math., vol. 19, iss. 3, pp. 306-381, 1976.
@article {P3, MRKEY = {0419491},
AUTHOR = {Procesi, C.},
TITLE = {The invariant theory of {$n\times n$} matrices},
JOURNAL = {Advances in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {19},
YEAR = {1976},
NUMBER = {3},
PAGES = {306--381},
ISSN = {0001-8708},
MRCLASS = {15A72 (14L99 20G20)},
MRNUMBER = {0419491},
MRREVIEWER = {M. Nagata},
ZBLNUMBER = {0331.15021},
} -
[Ri]
D. R. Richman, "The fundamental theorems of vector invariants," Adv. in Math., vol. 73, iss. 1, pp. 43-78, 1989.
@article {Ri, MRKEY = {0979587},
AUTHOR = {Richman, David R.},
TITLE = {The fundamental theorems of vector invariants},
JOURNAL = {Adv. in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {73},
YEAR = {1989},
NUMBER = {1},
PAGES = {43--78},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {20C07 (05A99 15A72)},
MRNUMBER = {0979587},
MRREVIEWER = {Andrea Brini},
DOI = {10.1016/0001-8708(89)90059-5},
ZBLNUMBER = {0668.15016},
} -
[RS]
H. Rui and M. Si, "A criterion on the semisimple Brauer algebras. II," J. Combin. Theory Ser. A, vol. 113, iss. 6, pp. 1199-1203, 2006.
@article {RS, MRKEY = {2244141},
AUTHOR = {Rui, Hebing and Si, Mei},
TITLE = {A criterion on the semisimple {B}rauer algebras. {II}},
JOURNAL = {J. Combin. Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {113},
YEAR = {2006},
NUMBER = {6},
PAGES = {1199--1203},
ISSN = {0097-3165},
CODEN = {JCBTA7},
MRCLASS = {16S99},
MRNUMBER = {2244141},
MRREVIEWER = {Carl Riehm},
DOI = {10.1016/j.jcta.2005.09.005},
ZBLNUMBER = {1108.16009},
} -
[W] H. Weyl, The Classical Groups, Princeton, NJ: Princeton Univ. Press, 1997.
@book {W, MRKEY = {1488158},
AUTHOR = {Weyl, Hermann},
TITLE = {The Classical Groups},
SERIES = {Princeton Landmarks in Math. Their Invariants and Representations},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1997},
PAGES = {xiv+320},
ISBN = {0-691-05756-7},
MRCLASS = {01A75 (20-03 20Cxx 20Gxx 22-01)},
MRNUMBER = {1488158},
ZBLNUMBER = {1024.20501},
} -
[X]
C. Xi, "On the quasi-heredity of Birman-Wenzl algebras," Adv. Math., vol. 154, iss. 2, pp. 280-298, 2000.
@article {X, MRKEY = {1784677},
AUTHOR = {Xi, Changchang},
TITLE = {On the quasi-heredity of {B}irman-{W}enzl algebras},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {154},
YEAR = {2000},
NUMBER = {2},
PAGES = {280--298},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {20C08 (16E10)},
MRNUMBER = {1784677},
MRREVIEWER = {Steffen K{ö}nig},
DOI = {10.1006/aima.2000.1919},
ZBLNUMBER = {0971.16008},
}